Problem: Integrate. $ \int 3\csc(x)\cot(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $3\csc^3(x)+3\csc(x)\cot^2(x) + C$ (Choice B) B $-3\csc^3(x)-3\csc(x)\cot^2(x) + C$ (Choice C) C $3\csc (x) + C$ (Choice D) D $-3\csc (x) + C$
Explanation: We need a function whose derivative is $3\csc(x)\cot(x)$. We know that the derivative of $\csc(x)$ is $-\csc(x)\cot(x)$, so let's start there: $\dfrac{d}{dx} \csc(x) = -\csc(x)\cot(x)$ Now let's multiply by $-3$ : $\dfrac{d}{dx}\left[ -3\csc(x)\right] = -3\dfrac{d}{dx}\csc(x) =3\csc(x)\cot(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 3\csc(x)\cot(x)\,dx =-3 \csc(x)\, + C$ The answer: $-3 \csc(x)\, + C$